1-Calculate-lim-x-0-tgx-m-sin-x-n-m-n-N-2-f-a-existe-calculate-lim-x-x-f-a-a-x-f-a-x-R-

Question Number 162726 by LEKOUMA last updated on 31/Dec/21

1) C a l c u l a t e

\lim_{x \to 0} \frac{{t g x}^{m}}{{(\sin x)}^{n}}, (m, n \in N)

2) f^{'} (a) e xiste, c a l c u l a t e

\lim_{x \to + \infty} x [f (a + \frac{a}{x}) - f (a - \frac{β}{x})],

(α, β \in R)

Answered by mr W last updated on 01/Jan/22

(1)

\lim_{x \to 0} \frac{{t g x}^{m}}{{(\sin x)}^{n}}

= \lim_{x \to 0} \frac{{t g x}^{m}}{x^{m}} \times {(\frac{x}{\sin x})}^{n} \times x^{m - n}

= \lim_{x \to 0} x^{m - n} = {\begin{cases} 1 i f m = n \\ 0 i f m > n \\ \infty i f m < n \end{cases}

(2)

\lim_{x \to + \infty} x [f (a + \frac{α}{x}) - f (a) - f (a - \frac{β}{x}) + f (a)]

= \lim_{x \to + \infty} [\frac{f (a + \frac{α}{x}) - f (a)}{\frac{1}{x}} - \frac{f (a - \frac{β}{x}) - f (a)}{\frac{1}{x}}]

= \lim_{x \to + \infty} [α \times \frac{f (a + \frac{α}{x}) - f (a)}{\frac{α}{x}} + β \times \frac{f (a - \frac{β}{x}) - f (a)}{- \frac{β}{x}}]

= \lim_{h, k \to 0} [α \times \frac{f (a + h) - f (a)}{h} + β \times \frac{f (a + k) - f (a)}{k}]

= α f^{'} (a) + β f^{'} (a)

= (α + β) f^{'} (a)

Commented by Ar Brandon last updated on 01/Jan/22

First comment of the year (GMT+1)
Happy New year, Sir !

Commented by mr W last updated on 01/Jan/22

t h a n k s!

t h e s a m e t o y o u a n d a l l o t h e r s!

Commented by Ar Brandon last updated on 01/Jan/22

Thanks Sir .